Theories of grammatical development differ in how much abstract knowledge they attribute to young children. Here, we report a series of experiments using a computational model to evaluate the explanatory power of child grammars based not on abstract rules but on concrete words and phrases and some local abstractions associated with these words and phrases. We use a Bayesian procedure to extract such item-based grammars from transcriptions of 28+ h of each of two children's speech at 2 and 3 years of age. We then use these grammars to parse all of the unique multiword utterances from transcriptions of separate recordings of these same children at each of the two ages. We found that at 2 years of age such a model had good coverage and predictive fit, with the children showing radically limited productivity. Furthermore, adding expert-annotated parts of speech to the induction procedure had little effect on coverage, with the exception of the category of noun. At age 3, the children's productivity sharply increased and the addition of a verb and a noun category markedly improved the model's performance.